Problem: Simplify the following expression: $ n = \dfrac{-1}{7} - \dfrac{8y - 7}{-10y - 7} $
In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{-10y - 7}{-10y - 7}$ $ \dfrac{-1}{7} \times \dfrac{-10y - 7}{-10y - 7} = \dfrac{10y + 7}{-70y - 49} $ Multiply the second expression by $\dfrac{7}{7}$ $ \dfrac{8y - 7}{-10y - 7} \times \dfrac{7}{7} = \dfrac{56y - 49}{-70y - 49} $ Therefore $ n = \dfrac{10y + 7}{-70y - 49} - \dfrac{56y - 49}{-70y - 49} $ Now the expressions have the same denominator we can simply subtract the numerators: $n = \dfrac{10y + 7 - (56y - 49) }{-70y - 49} $ Distribute the negative sign: $n = \dfrac{10y + 7 - 56y + 49}{-70y - 49}$ $n = \dfrac{-46y + 56}{-70y - 49}$ Simplify the expression by dividing the numerator and denominator by -1: $n = \dfrac{46y - 56}{70y + 49}$